• Pong
• Pac Man
• Tetris
• Robotron
• Space Invaders
• Lunar Lander
• Lode Runner

Two things that I consider important to consider:

1. Make something with a menu, high-scores, levels, and other things 'beyond the gameplay'.

2. Make something with a distinct hero-character that the player controls (i.e. not cursor based), as this tends to avoid many GUI-related side tracks.

Things that I consider important when choosing a game to remake:

1. Make a game that you have enjoyed playing in the past and are intimately familiar with. You want to know the design like the back of your hand so that you can focus on implimentation.

2. Make something with a menu, high-scores, levels, and other things 'beyond the gameplay'.

3. Make something with a distinct hero-character that the player controls (i.e. not cursor based), as this tends to avoid many GUI-related side tracks.

4. Pick a game for which you've previously imagined 'more modes' or other additional features, so that when you finish it you can immediately build upon it (if you choose).

The first point is really the thing. Asking for specific games is a bit over the top, anything here will get the job done really.

Edit: I had a list of games, focusing on advice instead....